Information Theory of Compressive Sensing in Sensor Array

Compressive Sensing (CS) can achieve a low number of software and hardware complexity by sampling at sub-Nyquist rates. This article proposes an information theory framework for the uniform linear array (ULA) systems applying CS technology to assess the compressive estimation performance. In this framework, we define Direction of Arrival (DOA) information and entropy error (EE) as forward and inverse indicators to evaluate the DOA estimation and derive theoretical expressions for DOA information and EE. Additionally, scattering information (SI) is presented for amplitude-phase performance assessment. Accordingly, we define the angle resolution limit (ARL) as the interval corresponding to the 1-bit quadrature component of SI. For closely spaced sources, we derive the closed-form approximation for ARL based on a second-order Taylor expansion. Theoretical analysis, relying on DOA information, EE, SI, and ARL, states that the performance degradation of the ULA system is significantly influenced by the compression ratio (CR). Simulation results demonstrate that for a given input signal-to-noise ratio, a smaller CR leads to diminished performance, characterized by lower DOA information and SI, and greater EE and ARL. Besides, the Cramér–Rao bound is derived for DOA estimation. Finally, the simulations are also conducted to illustrate the impact of various measurement matrices on the performance of the ULA systems. Consider partial identity, random Gaussian, random Bernoulli, partial Hadamard, and Toeplitz matrices. The results show that, except for the partial identity matrix, the other matrices fail to ensure uniform spatial spectrum distribution after compression, leading to unstable estimation performance. The proposed framework not only measures the compressive performance of the ULA systems but also provides a theoretical basis for further exploration based on this system.